I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example 1.2.1 (iii) on p.11.
From my understanding, the author is trying to derive a surface area measure on $S^2$ by a pushforward map. It starts with the fact that every compact group has a unique probability Haar measure on it. So is $\operatorname{SO}(3)=\{U\in M_3(\mathbb{R}):U^TU=I, \det U=1 \}$. It then says that the map $\varphi:SO(3)\rightarrow S^2$ given by $\varphi(U)=Ue_3$, where $e_3=(0,0,1)^T$ is the third canonical basis, induces a measure $\sigma_2$ on $S^2$, normalized to be a probability measure, from the Haar measure on $\operatorname{SO}(3)$.
Also, it mentions that since $\varphi$ is not bijective: $Ve_3=Ue_3$ i.f.f. $V^{-1}U\in\operatorname{SO}(2)\subset\operatorname{SO}(3)$. Thus we have that $\operatorname{SO}(3)/\operatorname{SO}(2)\cong S^2$.
Anyway, it gives the formula for the measure in terms of colatitude $\theta$ and longtitude $\phi$: $\mathrm{d}\sigma_2=\frac{1}{4\pi}\sin\theta\mathrm{d}\theta\mathrm{d}\phi$, which coincides with what I learned from advanced calculus course.
Can anyone tell me how he obtain this area measure? I want to know the procedure (as clear as possible) for computing this $d\sigma$. Thanks!
That's just the volume form of $S^2$ as a Riemannian submanifold of $\mathbb{R}^3$. You can compute it by taking the parametrization $$\Phi: \theta, \phi \mapsto (\cos(\theta)\sin(\phi), \sin(\theta)\sin(\phi), \cos(\phi))$$ and evaluating $$dx\wedge dy\wedge dz(\Phi_\ast\partial_{\theta}, \Phi_\ast\partial_{\phi}, n(\theta, \phi))$$ where n is the outward unit normal at $\Phi(\theta, \phi)$. Now since the Euclidean volume form is $SO(3,\mathbb{R})$ invariant, so is this spherical volume form. In general, if $G$ is compact, the homogeneous space $G/K$ will have a unique $G$ invariant probability measure. The basic idea is that functions on $G/K$ can be lifted to functions on $G$ and then integrated so that a Fubini formula obtains: $$\int_{G/K}\int_Kf(gk)dkd\dot{g}=\int_Gf(g)dg$$ So we've computed the unique $SO(3, \mathbb{R})$-invariant measure on $SO(3)/SO(2)=S^2$, up to normalization.
Maybe that answers your question?
If you want more abstraction/generality, you can identify $T_{e_3}(S^2)$ with the subspace $$ \left( \begin{array}{ccc} 0 & 0 & a \\ 0 & 0 & b \\ -a & -b & 0 \end{array} \right)$$ of the Lie algebra of $SO(3,\mathbb{R})$ via the derivative of your map $\varphi$. Then show that the dot product in $a$ and $b$ gives an $Ad(SO(2, \mathbb{R}))$ invariant inner product, which coincides with the Riemannian metric on $S^2$ induced from $\mathbb{R}^3$. Then show that your volume form is the one associated to this Riemannian metric.